In what follows we let \({H}\)  be a (separable) Hilbert space with inner product \(\left\langle \cdot  | \cdot  \right\rangle \) . We also denote by \({K}\)  the (closed) ideal of compact operators on \({H}\) . Given any operator \(T\in {K}\)  we let \(\mu =(\mu _j(T))_{j\ge 0}\)  be its sequence of singular values, i.e., \(\mu _j(T)\)  is the \((j+1)\) -the eigenvalue counted with multiplicity of the absolute value \(|T|=\sqrt{T^*T}\) . By the min-max principle [1]}, [2]} we have
\(\mu _j(T)&=\min \left\lbrace \Vert T_{|E^\perp }\Vert ;\ \dim E=j\right\rbrace , \\& = \min \left\lbrace \Vert T-R\Vert ;\ \operatorname{rk}(R)\le j\right\rbrace , \qquad j\ge 0.\) 
